Fuchsian tessellations in the hyperbolic plane have been considered by several authors (cf. [1], [2], [3]) to design signal constellations and nonuniform Fuchsian codes in the hyperbolic plane. They take advantage of strong relations with algebraic
structures, such as quaternion algebras and quadratic forms, and hyperbolic geometry.
We review the construction of those Fuchsian tessellations and explore its centers in order to obtain a good behaviour in the applications. The study of the centers blends euclidian and hyperbolic points of view, leading to consider the minimum border distance. Some examples of explicit constructions will
be given. A new approach is presented introducing bicenters and assignation based on probability of transmission.

The aim of the talk is to review the construction of fuchsian codes derived from orders in inde nite quaternion algebras and the associated fuchsian groups, published in recent papers as a joint work with I. Blanco-Chacon, C. Hollanti and D. Remon. Explicit examples allow to show how hyperbolic fundamental domains of fuchsian groups and the associated hyperbolic tessellations play an important role in the performance of these codes.
In this direction, we discuss why a careful study of centers of hyperbolic polygons would lead us to obtain properties on the codes, inorder to design better transmission schemes.

Signal constellations in the hyperbolic plane have been considered in several papers (cf. [2]-[5], among others), since hyperbolic geometry provides significative properties and strong relations with algebraic structures such as quaternion algebras, quadratic forms and Fuchsian groups (cf. [1]). In particular, infinitely many hyperbolic tessellations can be derived by using fundamental domains of Shimura curves associated to orders in quaternion algebras.
In this work we explore how to choose well centered points on those tessellations in order to obtain a good behaviour for applications to signals and codes. Hyperbolic geometry tools play an important role, and have been studied in conjunction with elements of euclidian geometry also present on the applications. Mathematics visualization software to deal with hyperbolic geometry is also explored.

Recently, Fuchsian codes have been proposed in Blanco-Chacon et al. (2014) [2] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1; e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.

Algebraic structures as Fuchsian groups have been applied to information theory to construct nonlinear codes. A new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups was presented at [1], and their generalization to higher rate was developped at [2]. In this talk we review the general construction and focus on the study of the center of those codes. In order to do that we use the geometry of the hyperbolic uniformizations of Shimura curves attached to the Fuchsian groups developped at [3]. This work is also related with a series of papers by Palazzo et al. a mong others (see for example [4]).

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.

The development of the European Higher Education Area (EHEA) scheme in Spanish universities implies a greater participation of the students in their own process of learning and competence-based teaching. Competences are the ability to apply knowledge, skills and attitudes and one of such
competences is communication. Students must be able to
communicate using their mother tongue, but they should be able to use a foreign language, especially English, too. This paper presents strategies applied in the School of Engineering at Manresa (EPSEM) (Barcelona), in order to improve students’ communication skills in English. An experimental research has been conducted, focusing on the point of view of engineering students, to check the efficacy of implementing content and
language integrated learning (CLIL). First, second and third year
students have been surveyed to test hypotheses about English level and implications on language and subject specific content learning. The data provide support to our hypotheses, showing an increasing positive attitude of the students towards studying subjects in English.

In order to ensure the quality of learning of Electronics for graduate studies offered at the EPSEM, it is needed to plan properly the subjects taught in English. In this paper the results of the students analysis who will attend these courses are shown. These results reveal the difficulties encountered and the
expectations generated. This analysis motives the development of specific support material.

We review the construction of Fuchsian groups from indefinite quaternion
algebras, giving explicit examples of the action of those groups on the complex upper half plane. Actually those groups are used to define the Modular Curves and the Shimura curves, which lie at the crossroads of many areas in mathematics and played a role in the proof of Shimura-Taniyama-Weil conjecture and the Last Fermat Theorem.
In the talk we explore how its hyperbolic uniformization is constructed
and lead us to a rich interpretation of the points in the complex upper halfplane.
By using this algebraic and geometric tools, a new application to
coding theory has been developed. Thus, from the derived hyperbolic tessellations on the complex plane, we will introduce the Fuchsian codes with some explicit examples.

The European Educational Institutions have the challenge and the commitment to enhance multilingual competence and teaching curricular subjects in a foreign language is seen as one of the most promising alternatives.
In that context, professors teaching different engineering subjects at the School of Engineering of the UPC at Manresa (EPSEM) have been involved in projects aiming at analyzing the current linguistic situation and developing some on-line open access materials using CLIL as a strategy. They formed the u-Linguatech Research Group on Multilingual Communication in Science and Technology in order to provide such resources in an effective and efficient way.
In this paper, we focus on students’ perception of the improvement of their
multilingual competence throughout their Engineering degree, by means of
subjects taught in English by non-native speakers. Data about the English level of current students are taken into account. We also describe the use of the above resources to improve the quality of subjects learning related to Chemical Engineering curricula.

In this paper, an optimal application of symbols is described, in the context of European Higher Education Area (EHEA) for engineering degrees. Handling mathematical symbols and foreign language are common competences, and the proposal is to deal with both at the same time.
On-line resources as Class-Talk help university lecturers and students to communicate more effectively in the classroom when a foreign language is used. In that setting the use of mathematics
symbols at engineering degrees highlighted the difficulties in the equivalence of verbal and symbolic languages. To manage them, some resources have been developed, mainly the application Multilingual Formulae presented in this paper.

A new transmission scheme for AWGN, based on Fuchsian groups, was proposed in [2], in such a way that maximal orders in indefinite quaternion algebra are used to generate the constellation.
We deal with embedding theory and quadratic forms to make explicit arithmetic fuchsian groups, specially those attached to small ramified quaternion algebras by using [1], in order to study the performance of derived codes.

Orders in indefinite quaternion algebras provide Fuchsian groups acting on the Poincare
half-plane, used to construct the associated Shimura curves.
We explain how, by using embedding theory, the elements of those Fuchsian groups depend
on representations of integers by suitable ternary quadratic forms. Thus the explicit computa-
tion of those representations leads to explicit presentations and fundamental domains of those
Fuchsian groups, the computation of CM points, and a rich interpretation of the points in the
complex upper half-plane.

Orders in indefinite quaternion algebras provide Fuchsian groups acting on the Poincare
half-plane, used to construct the associated Shimura curves.
We explain how, by using embedding theory, the elements of those Fuchsian groups depend
on representations of integers by suitable ternary quadratic forms. Thus the explicit computation
of those representations leads to explicit presentations and fundamental domains of those
Fuchsian groups, the computation of CM points, and a rich interpretation of the points in the
complex upper half-plane.

Indefinite quaternion algebras are used to define both arithmetic
Fuchsian groups and binary quadratic forms. We review these
definitions and show how the study of the action of Fuchsian
groups and their fundamental domains leads us to a rich interpretation
of the points in the complex upper half-plane.
In the talk we explore relationships between quaternion orders,
Shimura curves and quadratic forms, through complex
points. We get hyperbolic uniformizations of Shimura curves,
helpful in several ways: to give presentations of the groups, to
represent the special CM points, to define a general concept of
reduced binary quadratic forms, even if they have non integer
coefficients, etc.